// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2008 Gael Guennebaud <gael.guennebaud@inria.fr>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

#ifndef EIGEN_LLT_H
#define EIGEN_LLT_H

namespace Eigen {

namespace internal {

    template <typename _MatrixType, int _UpLo> struct traits<LLT<_MatrixType, _UpLo>> : traits<_MatrixType>
    {
        typedef MatrixXpr XprKind;
        typedef SolverStorage StorageKind;
        typedef int StorageIndex;
        enum
        {
            Flags = 0
        };
    };

    template <typename MatrixType, int UpLo> struct LLT_Traits;
}  // namespace internal

/** \ingroup Cholesky_Module
  *
  * \class LLT
  *
  * \brief Standard Cholesky decomposition (LL^T) of a matrix and associated features
  *
  * \tparam _MatrixType the type of the matrix of which we are computing the LL^T Cholesky decomposition
  * \tparam _UpLo the triangular part that will be used for the decompositon: Lower (default) or Upper.
  *               The other triangular part won't be read.
  *
  * This class performs a LL^T Cholesky decomposition of a symmetric, positive definite
  * matrix A such that A = LL^* = U^*U, where L is lower triangular.
  *
  * While the Cholesky decomposition is particularly useful to solve selfadjoint problems like  D^*D x = b,
  * for that purpose, we recommend the Cholesky decomposition without square root which is more stable
  * and even faster. Nevertheless, this standard Cholesky decomposition remains useful in many other
  * situations like generalised eigen problems with hermitian matrices.
  *
  * Remember that Cholesky decompositions are not rank-revealing. This LLT decomposition is only stable on positive definite matrices,
  * use LDLT instead for the semidefinite case. Also, do not use a Cholesky decomposition to determine whether a system of equations
  * has a solution.
  *
  * Example: \include LLT_example.cpp
  * Output: \verbinclude LLT_example.out
  *
  * \b Performance: for best performance, it is recommended to use a column-major storage format
  * with the Lower triangular part (the default), or, equivalently, a row-major storage format
  * with the Upper triangular part. Otherwise, you might get a 20% slowdown for the full factorization
  * step, and rank-updates can be up to 3 times slower.
  *
  * This class supports the \link InplaceDecomposition inplace decomposition \endlink mechanism.
  *
  * Note that during the decomposition, only the lower (or upper, as defined by _UpLo) triangular part of A is considered.
  * Therefore, the strict lower part does not have to store correct values.
  *
  * \sa MatrixBase::llt(), SelfAdjointView::llt(), class LDLT
  */
template <typename _MatrixType, int _UpLo> class LLT : public SolverBase<LLT<_MatrixType, _UpLo>>
{
public:
    typedef _MatrixType MatrixType;
    typedef SolverBase<LLT> Base;
    friend class SolverBase<LLT>;

    EIGEN_GENERIC_PUBLIC_INTERFACE(LLT)
    enum
    {
        MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
    };

    enum
    {
        PacketSize = internal::packet_traits<Scalar>::size,
        AlignmentMask = int(PacketSize) - 1,
        UpLo = _UpLo
    };

    typedef internal::LLT_Traits<MatrixType, UpLo> Traits;

    /**
      * \brief Default Constructor.
      *
      * The default constructor is useful in cases in which the user intends to
      * perform decompositions via LLT::compute(const MatrixType&).
      */
    LLT() : m_matrix(), m_isInitialized(false) {}

    /** \brief Default Constructor with memory preallocation
      *
      * Like the default constructor but with preallocation of the internal data
      * according to the specified problem \a size.
      * \sa LLT()
      */
    explicit LLT(Index size) : m_matrix(size, size), m_isInitialized(false) {}

    template <typename InputType> explicit LLT(const EigenBase<InputType>& matrix) : m_matrix(matrix.rows(), matrix.cols()), m_isInitialized(false)
    {
        compute(matrix.derived());
    }

    /** \brief Constructs a LLT factorization from a given matrix
      *
      * This overloaded constructor is provided for \link InplaceDecomposition inplace decomposition \endlink when
      * \c MatrixType is a Eigen::Ref.
      *
      * \sa LLT(const EigenBase&)
      */
    template <typename InputType> explicit LLT(EigenBase<InputType>& matrix) : m_matrix(matrix.derived()), m_isInitialized(false) { compute(matrix.derived()); }

    /** \returns a view of the upper triangular matrix U */
    inline typename Traits::MatrixU matrixU() const
    {
        eigen_assert(m_isInitialized && "LLT is not initialized.");
        return Traits::getU(m_matrix);
    }

    /** \returns a view of the lower triangular matrix L */
    inline typename Traits::MatrixL matrixL() const
    {
        eigen_assert(m_isInitialized && "LLT is not initialized.");
        return Traits::getL(m_matrix);
    }

#ifdef EIGEN_PARSED_BY_DOXYGEN
    /** \returns the solution x of \f$ A x = b \f$ using the current decomposition of A.
      *
      * Since this LLT class assumes anyway that the matrix A is invertible, the solution
      * theoretically exists and is unique regardless of b.
      *
      * Example: \include LLT_solve.cpp
      * Output: \verbinclude LLT_solve.out
      *
      * \sa solveInPlace(), MatrixBase::llt(), SelfAdjointView::llt()
      */
    template <typename Rhs> inline const Solve<LLT, Rhs> solve(const MatrixBase<Rhs>& b) const;
#endif

    template <typename Derived> void solveInPlace(const MatrixBase<Derived>& bAndX) const;

    template <typename InputType> LLT& compute(const EigenBase<InputType>& matrix);

    /** \returns an estimate of the reciprocal condition number of the matrix of
      *  which \c *this is the Cholesky decomposition.
      */
    RealScalar rcond() const
    {
        eigen_assert(m_isInitialized && "LLT is not initialized.");
        eigen_assert(m_info == Success && "LLT failed because matrix appears to be negative");
        return internal::rcond_estimate_helper(m_l1_norm, *this);
    }

    /** \returns the LLT decomposition matrix
      *
      * TODO: document the storage layout
      */
    inline const MatrixType& matrixLLT() const
    {
        eigen_assert(m_isInitialized && "LLT is not initialized.");
        return m_matrix;
    }

    MatrixType reconstructedMatrix() const;

    /** \brief Reports whether previous computation was successful.
      *
      * \returns \c Success if computation was successful,
      *          \c NumericalIssue if the matrix.appears not to be positive definite.
      */
    ComputationInfo info() const
    {
        eigen_assert(m_isInitialized && "LLT is not initialized.");
        return m_info;
    }

    /** \returns the adjoint of \c *this, that is, a const reference to the decomposition itself as the underlying matrix is self-adjoint.
      *
      * This method is provided for compatibility with other matrix decompositions, thus enabling generic code such as:
      * \code x = decomposition.adjoint().solve(b) \endcode
      */
    const LLT& adjoint() const EIGEN_NOEXCEPT { return *this; };

    inline EIGEN_CONSTEXPR Index rows() const EIGEN_NOEXCEPT { return m_matrix.rows(); }
    inline EIGEN_CONSTEXPR Index cols() const EIGEN_NOEXCEPT { return m_matrix.cols(); }

    template <typename VectorType> LLT& rankUpdate(const VectorType& vec, const RealScalar& sigma = 1);

#ifndef EIGEN_PARSED_BY_DOXYGEN
    template <typename RhsType, typename DstType> void _solve_impl(const RhsType& rhs, DstType& dst) const;

    template <bool Conjugate, typename RhsType, typename DstType> void _solve_impl_transposed(const RhsType& rhs, DstType& dst) const;
#endif

protected:
    static void check_template_parameters() { EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar); }

    /** \internal
      * Used to compute and store L
      * The strict upper part is not used and even not initialized.
      */
    MatrixType m_matrix;
    RealScalar m_l1_norm;
    bool m_isInitialized;
    ComputationInfo m_info;
};

namespace internal {

    template <typename Scalar, int UpLo> struct llt_inplace;

    template <typename MatrixType, typename VectorType>
    static Index llt_rank_update_lower(MatrixType& mat, const VectorType& vec, const typename MatrixType::RealScalar& sigma)
    {
        using std::sqrt;
        typedef typename MatrixType::Scalar Scalar;
        typedef typename MatrixType::RealScalar RealScalar;
        typedef typename MatrixType::ColXpr ColXpr;
        typedef typename internal::remove_all<ColXpr>::type ColXprCleaned;
        typedef typename ColXprCleaned::SegmentReturnType ColXprSegment;
        typedef Matrix<Scalar, Dynamic, 1> TempVectorType;
        typedef typename TempVectorType::SegmentReturnType TempVecSegment;

        Index n = mat.cols();
        eigen_assert(mat.rows() == n && vec.size() == n);

        TempVectorType temp;

        if (sigma > 0)
        {
            // This version is based on Givens rotations.
            // It is faster than the other one below, but only works for updates,
            // i.e., for sigma > 0
            temp = sqrt(sigma) * vec;

            for (Index i = 0; i < n; ++i)
            {
                JacobiRotation<Scalar> g;
                g.makeGivens(mat(i, i), -temp(i), &mat(i, i));

                Index rs = n - i - 1;
                if (rs > 0)
                {
                    ColXprSegment x(mat.col(i).tail(rs));
                    TempVecSegment y(temp.tail(rs));
                    apply_rotation_in_the_plane(x, y, g);
                }
            }
        }
        else
        {
            temp = vec;
            RealScalar beta = 1;
            for (Index j = 0; j < n; ++j)
            {
                RealScalar Ljj = numext::real(mat.coeff(j, j));
                RealScalar dj = numext::abs2(Ljj);
                Scalar wj = temp.coeff(j);
                RealScalar swj2 = sigma * numext::abs2(wj);
                RealScalar gamma = dj * beta + swj2;

                RealScalar x = dj + swj2 / beta;
                if (x <= RealScalar(0))
                    return j;
                RealScalar nLjj = sqrt(x);
                mat.coeffRef(j, j) = nLjj;
                beta += swj2 / dj;

                // Update the terms of L
                Index rs = n - j - 1;
                if (rs)
                {
                    temp.tail(rs) -= (wj / Ljj) * mat.col(j).tail(rs);
                    if (gamma != 0)
                        mat.col(j).tail(rs) = (nLjj / Ljj) * mat.col(j).tail(rs) + (nLjj * sigma * numext::conj(wj) / gamma) * temp.tail(rs);
                }
            }
        }
        return -1;
    }

    template <typename Scalar> struct llt_inplace<Scalar, Lower>
    {
        typedef typename NumTraits<Scalar>::Real RealScalar;
        template <typename MatrixType> static Index unblocked(MatrixType& mat)
        {
            using std::sqrt;

            eigen_assert(mat.rows() == mat.cols());
            const Index size = mat.rows();
            for (Index k = 0; k < size; ++k)
            {
                Index rs = size - k - 1;  // remaining size

                Block<MatrixType, Dynamic, 1> A21(mat, k + 1, k, rs, 1);
                Block<MatrixType, 1, Dynamic> A10(mat, k, 0, 1, k);
                Block<MatrixType, Dynamic, Dynamic> A20(mat, k + 1, 0, rs, k);

                RealScalar x = numext::real(mat.coeff(k, k));
                if (k > 0)
                    x -= A10.squaredNorm();
                if (x <= RealScalar(0))
                    return k;
                mat.coeffRef(k, k) = x = sqrt(x);
                if (k > 0 && rs > 0)
                    A21.noalias() -= A20 * A10.adjoint();
                if (rs > 0)
                    A21 /= x;
            }
            return -1;
        }

        template <typename MatrixType> static Index blocked(MatrixType& m)
        {
            eigen_assert(m.rows() == m.cols());
            Index size = m.rows();
            if (size < 32)
                return unblocked(m);

            Index blockSize = size / 8;
            blockSize = (blockSize / 16) * 16;
            blockSize = (std::min)((std::max)(blockSize, Index(8)), Index(128));

            for (Index k = 0; k < size; k += blockSize)
            {
                // partition the matrix:
                //       A00 |  -  |  -
                // lu  = A10 | A11 |  -
                //       A20 | A21 | A22
                Index bs = (std::min)(blockSize, size - k);
                Index rs = size - k - bs;
                Block<MatrixType, Dynamic, Dynamic> A11(m, k, k, bs, bs);
                Block<MatrixType, Dynamic, Dynamic> A21(m, k + bs, k, rs, bs);
                Block<MatrixType, Dynamic, Dynamic> A22(m, k + bs, k + bs, rs, rs);

                Index ret;
                if ((ret = unblocked(A11)) >= 0)
                    return k + ret;
                if (rs > 0)
                    A11.adjoint().template triangularView<Upper>().template solveInPlace<OnTheRight>(A21);
                if (rs > 0)
                    A22.template selfadjointView<Lower>().rankUpdate(A21, typename NumTraits<RealScalar>::Literal(-1));  // bottleneck
            }
            return -1;
        }

        template <typename MatrixType, typename VectorType> static Index rankUpdate(MatrixType& mat, const VectorType& vec, const RealScalar& sigma)
        {
            return Eigen::internal::llt_rank_update_lower(mat, vec, sigma);
        }
    };

    template <typename Scalar> struct llt_inplace<Scalar, Upper>
    {
        typedef typename NumTraits<Scalar>::Real RealScalar;

        template <typename MatrixType> static EIGEN_STRONG_INLINE Index unblocked(MatrixType& mat)
        {
            Transpose<MatrixType> matt(mat);
            return llt_inplace<Scalar, Lower>::unblocked(matt);
        }
        template <typename MatrixType> static EIGEN_STRONG_INLINE Index blocked(MatrixType& mat)
        {
            Transpose<MatrixType> matt(mat);
            return llt_inplace<Scalar, Lower>::blocked(matt);
        }
        template <typename MatrixType, typename VectorType> static Index rankUpdate(MatrixType& mat, const VectorType& vec, const RealScalar& sigma)
        {
            Transpose<MatrixType> matt(mat);
            return llt_inplace<Scalar, Lower>::rankUpdate(matt, vec.conjugate(), sigma);
        }
    };

    template <typename MatrixType> struct LLT_Traits<MatrixType, Lower>
    {
        typedef const TriangularView<const MatrixType, Lower> MatrixL;
        typedef const TriangularView<const typename MatrixType::AdjointReturnType, Upper> MatrixU;
        static inline MatrixL getL(const MatrixType& m) { return MatrixL(m); }
        static inline MatrixU getU(const MatrixType& m) { return MatrixU(m.adjoint()); }
        static bool inplace_decomposition(MatrixType& m) { return llt_inplace<typename MatrixType::Scalar, Lower>::blocked(m) == -1; }
    };

    template <typename MatrixType> struct LLT_Traits<MatrixType, Upper>
    {
        typedef const TriangularView<const typename MatrixType::AdjointReturnType, Lower> MatrixL;
        typedef const TriangularView<const MatrixType, Upper> MatrixU;
        static inline MatrixL getL(const MatrixType& m) { return MatrixL(m.adjoint()); }
        static inline MatrixU getU(const MatrixType& m) { return MatrixU(m); }
        static bool inplace_decomposition(MatrixType& m) { return llt_inplace<typename MatrixType::Scalar, Upper>::blocked(m) == -1; }
    };

}  // end namespace internal

/** Computes / recomputes the Cholesky decomposition A = LL^* = U^*U of \a matrix
  *
  * \returns a reference to *this
  *
  * Example: \include TutorialLinAlgComputeTwice.cpp
  * Output: \verbinclude TutorialLinAlgComputeTwice.out
  */
template <typename MatrixType, int _UpLo> template <typename InputType> LLT<MatrixType, _UpLo>& LLT<MatrixType, _UpLo>::compute(const EigenBase<InputType>& a)
{
    check_template_parameters();

    eigen_assert(a.rows() == a.cols());
    const Index size = a.rows();
    m_matrix.resize(size, size);
    if (!internal::is_same_dense(m_matrix, a.derived()))
        m_matrix = a.derived();

    // Compute matrix L1 norm = max abs column sum.
    m_l1_norm = RealScalar(0);
    // TODO move this code to SelfAdjointView
    for (Index col = 0; col < size; ++col)
    {
        RealScalar abs_col_sum;
        if (_UpLo == Lower)
            abs_col_sum = m_matrix.col(col).tail(size - col).template lpNorm<1>() + m_matrix.row(col).head(col).template lpNorm<1>();
        else
            abs_col_sum = m_matrix.col(col).head(col).template lpNorm<1>() + m_matrix.row(col).tail(size - col).template lpNorm<1>();
        if (abs_col_sum > m_l1_norm)
            m_l1_norm = abs_col_sum;
    }

    m_isInitialized = true;
    bool ok = Traits::inplace_decomposition(m_matrix);
    m_info = ok ? Success : NumericalIssue;

    return *this;
}

/** Performs a rank one update (or dowdate) of the current decomposition.
  * If A = LL^* before the rank one update,
  * then after it we have LL^* = A + sigma * v v^* where \a v must be a vector
  * of same dimension.
  */
template <typename _MatrixType, int _UpLo>
template <typename VectorType>
LLT<_MatrixType, _UpLo>& LLT<_MatrixType, _UpLo>::rankUpdate(const VectorType& v, const RealScalar& sigma)
{
    EIGEN_STATIC_ASSERT_VECTOR_ONLY(VectorType);
    eigen_assert(v.size() == m_matrix.cols());
    eigen_assert(m_isInitialized);
    if (internal::llt_inplace<typename MatrixType::Scalar, UpLo>::rankUpdate(m_matrix, v, sigma) >= 0)
        m_info = NumericalIssue;
    else
        m_info = Success;

    return *this;
}

#ifndef EIGEN_PARSED_BY_DOXYGEN
template <typename _MatrixType, int _UpLo>
template <typename RhsType, typename DstType>
void LLT<_MatrixType, _UpLo>::_solve_impl(const RhsType& rhs, DstType& dst) const
{
    _solve_impl_transposed<true>(rhs, dst);
}

template <typename _MatrixType, int _UpLo>
template <bool Conjugate, typename RhsType, typename DstType>
void LLT<_MatrixType, _UpLo>::_solve_impl_transposed(const RhsType& rhs, DstType& dst) const
{
    dst = rhs;

    matrixL().template conjugateIf<!Conjugate>().solveInPlace(dst);
    matrixU().template conjugateIf<!Conjugate>().solveInPlace(dst);
}
#endif

/** \internal use x = llt_object.solve(x);
  *
  * This is the \em in-place version of solve().
  *
  * \param bAndX represents both the right-hand side matrix b and result x.
  *
  * This version avoids a copy when the right hand side matrix b is not needed anymore.
  *
  * \warning The parameter is only marked 'const' to make the C++ compiler accept a temporary expression here.
  * This function will const_cast it, so constness isn't honored here.
  *
  * \sa LLT::solve(), MatrixBase::llt()
  */
template <typename MatrixType, int _UpLo> template <typename Derived> void LLT<MatrixType, _UpLo>::solveInPlace(const MatrixBase<Derived>& bAndX) const
{
    eigen_assert(m_isInitialized && "LLT is not initialized.");
    eigen_assert(m_matrix.rows() == bAndX.rows());
    matrixL().solveInPlace(bAndX);
    matrixU().solveInPlace(bAndX);
}

/** \returns the matrix represented by the decomposition,
 * i.e., it returns the product: L L^*.
 * This function is provided for debug purpose. */
template <typename MatrixType, int _UpLo> MatrixType LLT<MatrixType, _UpLo>::reconstructedMatrix() const
{
    eigen_assert(m_isInitialized && "LLT is not initialized.");
    return matrixL() * matrixL().adjoint().toDenseMatrix();
}

/** \cholesky_module
  * \returns the LLT decomposition of \c *this
  * \sa SelfAdjointView::llt()
  */
template <typename Derived> inline const LLT<typename MatrixBase<Derived>::PlainObject> MatrixBase<Derived>::llt() const { return LLT<PlainObject>(derived()); }

/** \cholesky_module
  * \returns the LLT decomposition of \c *this
  * \sa SelfAdjointView::llt()
  */
template <typename MatrixType, unsigned int UpLo>
inline const LLT<typename SelfAdjointView<MatrixType, UpLo>::PlainObject, UpLo> SelfAdjointView<MatrixType, UpLo>::llt() const
{
    return LLT<PlainObject, UpLo>(m_matrix);
}

}  // end namespace Eigen

#endif  // EIGEN_LLT_H
